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The Center for Bright Beams, A National Science Foundation Science and Technology Center

Role of surface defects and material inhomogeneities

Role of surface defects and material inhomogeneities for vortex nucleation in superconductors within time-dependent Ginzburg-Landau theory in 2 and 3 dimensions

Here we plot the norm squared of the superconducting order parameter on a 3D film. The norm squared of the order parameter is a measure of how superconducting the material is, zero being nonsuperconducting and 1 being the typical superconducting state. In this plot we see that a larger applied magnetic field (Ha) is required to nucleate vortices if there is a divot on the surface perpendicular to the direction of the applied magnetic field.

We use Time-Dependent Ginzburg-Landau theory to study the nucleation of vortices in type II superconductors in the presence of both geometric and material inhomogeneities. The superconducting Meissner state is meta-stable up to a critical magnetic field, known as the superheating field (Hsh). For a uniform surface and homogenous material, the superheating transition is driven by a non-local critical mode in which an array of vortices simultaneously penetrate the surface. In contrast, we show that even a small amount of disorder localizes the critical mode and can have a significant reduction in the effective superheating field for a particular sample. Vortices can be nucleated by either surface roughness or local variations in material parameters, such as Tc. Our approach uses a finite element method to simulate a cylindrical geometry in 2 dimensions and a film geometry in 2 and 3 dimensions. We combine saddle node bifurcation analysis along with a novel fitting procedure to evaluate the superheating field and identify the unstable mode. We demonstrate agreement with previous results for homogenous geometries and surface roughness and extend the analysis to include variations in material properties. Finally, we show that in three dimensions, surface divots not aligned with the applied field can increase the super heating field. We discuss implications for fabrication and performance of superconducting resonant frequency (SRF) cavities in particle accelerators.

Applications and Relation to CBB Goals:

Nb₃Sn SRF cavities used to accelerate particles close to the speed of light sometimes contain defects such as islands of depleted Sn, grain boundaries, and surface roughness. It is very difficult to measure how these defects impact vortex nucleation and reduce the maximum magnetic field they can operate at. This article demonstrates how we use computers to simulate vortex nucleation and how to calculate Hsh in Time-Dependent Ginzburg-Landau theory. This foundational work sets the stage for future calculations where we show the impact of material specific defects on Nb₃Sn SRF cavity performance. By informing CBB and the accelerator community which defects are the most detrimental we hope to provide insight into potential changes in Nb₃Sn fabrication that can lead to higher accelerating gradients and brighter beams.


A. R. Pack, J. Carlson, S. Wadsworth, and M. K. Transtrum, “Role of surface defects and material inhomogeneities for vortex nucleation in superconductors within time-dependent Ginzburg-Landau theory in 2 and 3 dimensions,” Phys. Rev. B, vol. In press., Feb. 2020 [Online]. Available: